nLab inductive-recursive type



Deduction and Induction

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels





In type theory, induction-recursion is a principle for mutually defining types of the form

Atypea:AB(a)typeA \; \mathrm{type} \qquad a:A \vdash B(a) \; \mathrm{type}

where AA is defined as an inductive type and BB is defined by recursion on AA. Crucially, the definition of AA may use BB. Without this last requirement, we could first define AA and then separately BB.

In type theory, higher induction-reduction is the principle that one could also have identity constructors and identity section constructors in addition to the element and section constructors.


A Tarski universe is an example of an inductive-recursive definition, where a type UU is defined inductively together with a type family a:UT(a)typea:U \vdash T(a) \; \mathrm{type}. The constructors for UU may depend negatively on TT applied to elements of UU. This is the case if UU, for example, is closed under dependent product types, where it has constructors of

  • a dependent type Π(a,b):U\Pi(a, b):U for each a:Ua:U and b:T(a)Ub:T(a) \to U,
  • a function E Π(a,b):( x:T(a)T(b(x)))T(Π(a,b))E_\Pi(a, b):\left(\prod_{x:T(a)} T(b(x))\right) \to T(\Pi(a, b)) for each a:Ua:U and b:T(a)Ub:T(a) \to U.

If the Tarski universe is strictly closed under dependent product types, then the last condition is replaced by

  • a judgmental equality T(Π(a,b)) x:T(a)T(b(x))T(\Pi(a, b)) \equiv \prod_{x:T(a)} T(b(x)) for each a:Ua:U and b:T(a)Ub:T(a) \to U.

Here, the type family TT is defined recursively. Sometimes, however, one might not want to give T(u)T(u) completely as soon as u:Uu:U is introduced, but instead define TT inductively as well. This is the principle of induction-induction.


History of inductive types

Historical references on the definition of inductive types.


A first type theoretic formulation of general inductive definitions:

The induction principle for identity types (also known as “path induction” or the “J-rule”) is first stated in:

  • Per Martin-Löf, §1.7 and p. 94 of: An intuitionistic theory of types: predicative part, in: H. E. Rose, J. C. Shepherdson (eds.), Logic Colloquium ‘73, Proceedings of the Logic Colloquium, Studies in Logic and the Foundations of Mathematics 80 (1975) 73-118 (doi:10.1016/S0049-237X(08)71945-1, CiteSeer)

and in the modern form of inference rules in:

  • Bengt Nordström, Kent Petersson, Jan M. Smith, §8.1 of: Programming in Martin-Löf’s Type Theory, Oxford University Press (1990) [[webpage, pdf, pdf]]

The special case of inductive types now known as 𝒲 \mathcal{W} -types is first formulated in:

Early proposals for a general formal definition of inductive types:

Modern definition

The modern notion of inductive types and inductive families in intensional type theory is independently due to

and due to

which became the basis of the calculus of inductive constructions used in the Coq-proof assistant:

reviewed in

with streamlined exposition in:

The generalization to inductive-recursive types is due to

See also:

Further development


Last revised on October 5, 2023 at 13:38:51. See the history of this page for a list of all contributions to it.